Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Wallman-type compactifications on 0-dimensional spaces


Author: Li Pi Su
Journal: Proc. Amer. Math. Soc. 43 (1974), 455-460
MSC: Primary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1974-0339079-9
MathSciNet review: 0339079
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E$ be Hausdorff 0-dimensional, $ \mathcal{D}$ the discrete space $ \{0, 1\}$, and $ \mathcal{N}$ the discrete space of all nonnegative integers. Every $ E$-completely regular space $ X$ has a clopen normal base $ \mathcal{F}$ with $ X\backslash F \in \mathcal{F}$ for each $ F \in \mathcal{F}$. The Wallman compactification $ \omega (\mathcal{F})$ is $ \mathcal{D}$-compact. Moreover, if an $ E$-completely regular space $ X$ has a countably productive clopen normal base $ \mathcal{F}$ with $ X\backslash F \in \mathcal{F}$ for each $ F \in \mathcal{F}$, then the Wallman space $ \eta (\mathcal{F})$ is $ \mathcal{N}$-compact. Hence, if $ X$ has such an $ \mathcal{F}$, and is an $ \mathcal{F}$-realcompact space, then $ X$ is $ \mathcal{N}$-compact.


References [Enhancements On Off] (What's this?)

  • [1] R. A. Alo and H. L. Shapiro, A note on compactifications and seminormal spaces, J. Austral. Math. Soc. 8 (1968), 102-108. MR 37 #3527. MR 0227943 (37:3527)
  • [2] -, Normal bases and compactifications. Math. Ann. 175 (1968), 337-340. MR 36 #3312. MR 0220246 (36:3312)
  • [3] -, Wallman compact realcompact spaces, Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967), Deutsch. Verlag Wissen., Berlin, 1969, pp. 9-14. MR 40 #872. MR 0247608 (40:872)
  • [4] -, $ \mathcal{Z}$-real compactifications and normal bases, J. Austral. Math. Soc. 9 (1969), 489-495. MR 39 #3455. MR 0242121 (39:3455)
  • [5] K. P. Chew, A characterization of $ \mathcal{N}$-compact spaces, Proc. Amer. Math. Soc. 26 (1970), 679-682. MR 42 #2436. MR 0267534 (42:2436)
  • [6] O. Frink, Compactification and semi-normal spaces, Amer. J. Math. 86 (1964), 602-607. MR 29 #4028. MR 0166755 (29:4028)
  • [7] Z. Frolík, A generalization of realcompact spaces, Czechoslovak Math. J. 13 (88) (1963), 127-138. MR 27 #5224. MR 0155289 (27:5224)
  • [8] S. Mrówka, Further results on $ E$-compact spaces. I, Acta. Math. 120 (1968), 161-185. MR 37 #2165. MR 0226576 (37:2165)
  • [9] O. Njastad, On Wallman-type compactifications, Math. Z. 91 (1966), 267-276. MR 32 #6404. MR 0188977 (32:6404)
  • [10] N. Piacun and L. P. Su, Wallman compactifications on $ E$-completely regular spaces, Pacific J. Math. 45 (1973), 321-326. MR 0319148 (47:7694)
  • [11] E. F. Steiner, Wallman spaces and compactifications, Fund. Math. 61 (1967/68), 295-304. MR 36 #5899. MR 0222849 (36:5899)
  • [12] A. K. Steiner and E. F. Steiner, Nest generated intersection rings in Tychonoff spaces, Trans. Amer. Math. Soc. 148 (1970), 589-601. MR 41 #7637. MR 0263032 (41:7637)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D35

Retrieve articles in all journals with MSC: 54D35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0339079-9
Keywords: Wallman spaces, Wallman compactifications, $ E$-completely regular, $ \mathcal{D}$-compact, $ \mathcal{N}$-compact, 0-dimension, complemental base
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society